1. The problem is to express $8^4$ with base 2. 2. Recall that $8$ can be written as a power of 2: $8 = 2^3$. 3. Using the exponentiation rule $(a^m)^n = a^{m \times n}$, we rewrite $8^4$ as $(2^3)^4$. 4. Applying the rule, we get $2^{3 \times 4} = 2^{12}$. 5. Therefore, $8^4$ expressed with base 2 is $2^{12}$.